Chapter 5.7, Problem 8RE

a.

To determine

Check whether the statement “If you add 7 to each entry on a list that adds 7 to the average” is true or false explain briefly.

Answer to Problem 8RE

The given statement is true.

Explanation of Solution

In general, adding 7 to each list entry would add 7 to the average number.

For example, consider the list as 1, 2, and 1.

The average of the list is obtained as follows:

$\begin{array}{c}\text{Average=}\frac{\text{Sum of data values}}{\text{Total number of data values}}\\ =\frac{1+2+1}{3}\\ =1.33\end{array}$

Thus, the average of the data is 1.33.

Add 7 to the list of 1, 2 and 1.

The list of entries after adding 7 is 8, 9, and 8.

The average for the new list is shown below:

$\begin{array}{c}\text{Average=}\frac{\text{Sum of data values}}{\text{Total number of data values}}\\ =\frac{8+9+8}{3}\\ =8.33\end{array}$

Thus, the average of the new list is 8.33.

Hence, adding 7 to each list entry would add 7 to the average number.

Therefore, the statement “If you add 7 to each entry on a list that adds 7 to the average” is true.

b.

To determine

Check whether the statement “If you add 7 to each entry on a list that adds 7 to the SD” is true or false explain briefly.

Answer to Problem 8RE

The given statement is false.

Explanation of Solution

Here, the deviations from the average won’t change. Thus, the SD remains constant when add 7 to each entry on a list.

For example, consider the list as 1, 2, and 1.

From Part (a), the average of 1, 2 and 1 is 1.33.

The standard deviation of the list is obtained as follows:

$\begin{array}{c}SD=\sqrt{\frac{\text{Sum of }\left({\left(\text{Entry}-\text{Average}\right)}^2\right)}{\text{Number of data values }-1}}\\ =\sqrt{\frac{{\left(1-1.33\right)}^2+{\left(2-1.33\right)}^2+{\left(1-1.33\right)}^2}{3-1}}\\ =0.5774\end{array}$

Thus, the average of the data is 0.5774.

Add 7 to the list of 1, 2 and 1.

The list of entries after adding 7 is 8, 9, and 8. From Part (a), the average of the new list is 8.33.

$\begin{array}{c}SD=\sqrt{\frac{\text{Sum of }\left({\left(\text{Entry}-\text{Average}\right)}^2\right)}{\text{Number of data values }-1}}\\ =\sqrt{\frac{{\left(8-8.33\right)}^2+{\left(9-8.33\right)}^2+{\left(8-8.33\right)}^2}{3-1}}\\ =0.5774\end{array}$

Thus, the SD for the new list is 0.5774.

Here, the SD remains constant when add 7 to each entry on a list.

Thus, the statement “If you add 7 to each entry on a list that adds 7 to the SD” is false.

c.

To determine

Check whether the statement “If you double each entry on a list that doubles the average” is true or false explain briefly.

Answer to Problem 8RE

The given statement is true.

Explanation of Solution

If every element in a list is doubled, the average is doubled.

For example, consider the list as 1, 2, and 1.

From Part (a), the average is 1.33.

Double each entry in the list. Therefore, the new list is 2, 4, and 2.

The average for the new list is obtained as follows:

$\begin{array}{c}\text{Average=}\frac{\text{Sum of data values}}{\text{Total number of data values}}\\ =\frac{2+4+2}{3}\\ =2.66\end{array}$

Here, the average of the list is doubled. Therefore, the given statement is true.

d.

To determine

Check whether the statement “If you double each entry on a list that doubles the SD” is true or false explain briefly.

Answer to Problem 8RE

The given statement is true.

Explanation of Solution

Here, the deviations from the average are doubled. Thus, if every element in a list is doubled, the SD is doubled.

For example, consider the list as 1, 2, and 1.

From Part (c), the new average for 2, 4 and 2 is 2.66.

From Part (b), the SD is 0.5775.

Double each entry in the list. Therefore, the new list is 2, 4, and 2.

The SD for the new list is obtained as follows:

$\begin{array}{c}SD=\sqrt{\frac{\text{Sum of }\left({\left(\text{Entry}-\text{Average}\right)}^2\right)}{\text{Number of data values }-1}}\\ =\sqrt{\frac{{\left(2-2.66\right)}^2+{\left(4-2.66\right)}^2+{\left(2-2.66\right)}^2}{3-1}}\\ =1.155\end{array}$

Here, the SD of the list is doubled. Therefore, the given statement is true.

e.

To determine

Check whether the statement “If you change the sign of each entry on a list, that changes the sign of the average” is true or false explain briefly.

Answer to Problem 8RE

The given statement is true.

Explanation of Solution

Here, if each entry in a list changes the sign, this changes the average sign.

For example, consider the list as 1, 2, and 1.

From Part (a), the average is 1.33.

The new list of the entries after changing the sign is –1, –2 and –1.

The average for the new list is shown below:

$\begin{array}{c}\text{Average=}\frac{\text{Sum of data values}}{\text{Total number of data values}}\\ =\frac{-1-2-1}{3}\\ =-1.33\end{array}$

Thus, the average of the new list is –1.33.

Hence, the statement “If you change the sign of each entry on a list, that changes the sign of the average” is true

f.

To determine

Check whether the statement “If you change the sign of each entry on a list, that changes the sign of the SD” is true or false explain briefly.

Answer to Problem 8RE

The given statement is false.

Explanation of Solution

Here, if each entry in a list changes the sign, this does not change the SD sign.

For example, consider the list as 1, 2, and 1.

From Part (b), the SD is 0.5774.

The new list of the entries after changing the sign is –1, –2 and –1.

The average for the new list is shown below:

$\begin{array}{c}SD=\sqrt{\frac{\text{Sum of }\left({\left(\text{Entry}-\text{Average}\right)}^2\right)}{\text{Number of data values }-1}}\\ =\sqrt{\frac{{\left(-1+1.33\right)}^2+{\left(-2+1.33\right)}^2+{\left(-1+1.33\right)}^2}{3-1}}\\ =0.5774\end{array}$

Thus, the SD is 0.5774.

Here, the sign does not change the SD.

Hence, the statement “If you change the sign of each entry on a list that changes the sign of the SD” is false.